Bernoulli

  • Bernoulli
  • Volume 24, Number 4A (2018), 3039-3086.

Nested particle filters for online parameter estimation in discrete-time state-space Markov models

Dan Crisan and Joaquín Míguez

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Abstract

We address the problem of approximating the posterior probability distribution of the fixed parameters of a state-space dynamical system using a sequential Monte Carlo method. The proposed approach relies on a nested structure that employs two layers of particle filters to approximate the posterior probability measure of the static parameters and the dynamic state variables of the system of interest, in a vein similar to the recent “sequential Monte Carlo square” (SMC$^{2}$) algorithm. However, unlike the SMC$^{2}$ scheme, the proposed technique operates in a purely recursive manner. In particular, the computational complexity of the recursive steps of the method introduced herein is constant over time. We analyse the approximation of integrals of real bounded functions with respect to the posterior distribution of the system parameters computed via the proposed scheme. As a result, we prove, under regularity assumptions, that the approximation errors vanish asymptotically in $L_{p}$ ($p\ge1$) with convergence rate proportional to $\frac{1}{\sqrt{N}}+\frac{1}{\sqrt{M}}$, where $N$ is the number of Monte Carlo samples in the parameter space and $N\times M$ is the number of samples in the state space. This result also holds for the approximation of the joint posterior distribution of the parameters and the state variables. We discuss the relationship between the SMC$^{2}$ algorithm and the new recursive method and present a simple example in order to illustrate some of the theoretical findings with computer simulations.

Article information

Source
Bernoulli, Volume 24, Number 4A (2018), 3039-3086.

Dates
Received: August 2015
Revised: October 2016
First available in Project Euclid: 26 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.bj/1522051233

Digital Object Identifier
doi:10.3150/17-BEJ954

Mathematical Reviews number (MathSciNet)
MR3779710

Zentralblatt MATH identifier
06853273

Keywords
error bounds model inference Monte Carlo parameter estimation particle filtering recursive algorithms state space models

Citation

Crisan, Dan; Míguez, Joaquín. Nested particle filters for online parameter estimation in discrete-time state-space Markov models. Bernoulli 24 (2018), no. 4A, 3039--3086. doi:10.3150/17-BEJ954. https://projecteuclid.org/euclid.bj/1522051233


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References

  • [1] Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B Stat. Methodol. 72 269–342.
  • [2] Andrieu, C., Doucet, A., Singh, S.S. and Tadić, V.B. (2004). Particle methods for change detection, system identification and control. Proc. IEEE 92 423–438.
  • [3] Beskos, A., Crisan, D. and Jasra, A. (2014). On the stability of sequential Monte Carlo methods in high dimensions. Ann. Appl. Probab. 24 1396–1445.
  • [4] Bruno, M.G.S. (2013). Sequential Monte Carlo methods for nonlinear discrete-time filtering. Synthesis Lectures on Signal Processing 6 1–99.
  • [5] Cappé, O., Godsill, S.J. and Moulines, E. (2007). An overview of existing methods and recent advances in sequential Monte Carlo. Proc. IEEE 95 899–924.
  • [6] Cappé, O., Guillin, A., Marin, J.M. and Robert, C.P. (2004). Population Monte Carlo. J. Comput. Graph. Statist. 13 907–929.
  • [7] Carvalho, C.M., Johannes, M.S., Lopes, H.F. and Polson, N.G. (2010). Particle learning and smoothing. Statist. Sci. 25 88–106.
  • [8] Chen, R., Wang, X. and Liu, J.S. (2000). Adaptive joint detection and decoding in flat-fading channels via mixture Kalman filtering. IEEE Trans. Inform. Theory 46 2079–2094.
  • [9] Chopin, N. (2002). A sequential particle filter method for static models. Biometrika 89 539–551.
  • [10] Chopin, N., Jacob, P.E. and Papaspiliopoulos, O. (2013). $\mathrm{SMC}^{2}$: An efficient algorithm for sequential analysis of state space models. J. R. Stat. Soc. Ser. B. Stat. Methodol. 75 397–426.
  • [11] Chorin, A.J. and Krause, P. (2004). Dimensional reduction for a Bayesian filter. Proc. Natl. Acad. Sci. USA 101 15013–15017.
  • [12] Crisan, D. (2001). Particle filters—a theoretical perspective. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.). Stat. Eng. Inf. Sci. 17–41. New York: Springer.
  • [13] Crisan, D. and Doucet, A. (2002). A survey of convergence results on particle filtering methods for practitioners. IEEE Trans. Signal Process. 50 736–746.
  • [14] Del Moral, P. (2004). Feynman–Kac Formulae: Genealogical and Interacting Particle Systems with Applications. Probability and Its Applications (New York). New York: Springer.
  • [15] Del Moral, P., Doucet, A. and Singh, S.S. (2015). Uniform stability of a particle approximation of the optimal filter derivative. SIAM J. Control Optim. 53 1278–1304.
  • [16] Del Moral, P. and Miclo, L. (2000). Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear filtering. In Séminaire de Probabilités, XXXIV. Lecture Notes in Math. 1729 1–145. Berlin: Springer.
  • [17] Douc, R., Cappé, O. and Moulines, E. (2005). Comparison of resampling schemes for particle filtering. In Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis 64–69.
  • [18] Doucet, A., de Freitas, N. and Gordon, N. (2001). An introduction to sequential Monte Carlo methods. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.). Stat. Eng. Inf. Sci. 3–14. New York: Springer.
  • [19] Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo Sampling methods for Bayesian filtering. Stat. Comput. 10 197–208.
  • [20] Fearnhead, P. (2002). Markov chain Monte Carlo, sufficient statistics, and particle filters. J. Comput. Graph. Statist. 11 848–862.
  • [21] Gilks, W.R. and Berzuini, C. (2001). Following a moving target—Monte Carlo inference for dynamic Bayesian models. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 127–146.
  • [22] Gordon, N., Salmond, D. and Smith, A.F.M. (1993). Novel approach to nonlinear and non-Gaussian Bayesian state estimation. IEE Proc. F 140 107–113.
  • [23] Kantas, N., Doucet, A., Singh, S.S., Maciejowski, J. and Chopin, N. (2015). On particle methods for parameter estimation in state-space models. Statist. Sci. 30 328–351.
  • [24] Kitagawa, G. (1996). Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graph. Statist. 5 1–25.
  • [25] Kitagawa, G. (1998). A self-organizing state-space model. J. Amer. Statist. Assoc. 1203–1215.
  • [26] Koblents, E. and Míguez, J. (2013). A population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models. Stat. Comput.
  • [27] Koblents, E. and Míguez, J. (2015). A population Monte Carlo scheme with transformed weights and its application to stochastic kinetic models. Stat. Comput. 25 407–425.
  • [28] Kong, A., Liu, J.S. and Wong, W.H. (1994). Sequential imputations and Bayesian missing data problems. J. Amer. Statist. Assoc. 9 278–288.
  • [29] LeGland, F. and Mevel, L. (1997). Recursive estimation in hidden Markov models. In Proceedings of the 36th IEEE Conference on Decision and Control 4 3468–3473. IEEE.
  • [30] Liu, J. and West, M. (2001). Combined parameter and state estimation in simulation-based filtering. In Sequential Monte Carlo Methods in Practice (A. Doucet, N. de Freitas and N. Gordon, eds.). Stat. Eng. Inf. Sci. 197–223. New York: Springer.
  • [31] Liu, J.S. and Chen, R. (1998). Sequential Monte Carlo methods for dynamic systems. J. Amer. Statist. Assoc. 93 1032–1044.
  • [32] Lorenz, E.N. (1963). Deterministic nonperiodic flow. J. Atmos. Sci. 20 130–141.
  • [33] Maíz, C.S., Molanes-López, E.M., Míguez, J. and Djurić, P.M. (2012). A particle filtering scheme for processing time series corrupted by outliers. IEEE Trans. Signal Process. 60 4611–4627.
  • [34] Miguez, J., Bugallo, M. and Djuric, P.M. (2005). Novel particle filtering algorithms for fixed parameter estimation in dynamic systems. In Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis (ISPA) 46–51. IEEE.
  • [35] Míguez, J., Crisan, D. and Djurić, P.M. (2013). On the convergence of two sequential Monte Carlo methods for maximum a posteriori sequence estimation and stochastic global optimization. Stat. Comput. 23 91–107.
  • [36] Olsson, J., Cappé, O., Douc, R. and Moulines, E. (2008). Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models. Bernoulli 14 155–179.
  • [37] Papavasiliou, A. (2006). Parameter estimation and asymptotic stability in stochastic filtering. Stochastic Process. Appl. 116 1048–1065.
  • [38] Poyiadjis, G., Doucet, A. and Singh, S.S. (2011). Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika 98 65–80.
  • [39] Ristic, B., Arulampalam, S. and Gordon, N. (2004). Beyond the Kalman Filter: Particle Filters for Tracking Applications. Boston: Artech House.
  • [40] Storvik, G. (2002). Particle filters for state-space models with the presence of unknown static parameters. IEEE Trans. Signal Process. 50 281–289.